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APPROXIMATE AND LOW REGULARITY DIRICHLET Victor.Nistor/ART/gfdd.pdf · PDF...

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  • APPROXIMATE AND LOW REGULARITY DIRICHLETBOUNDARY CONDITIONS IN THE GENERALIZED FINITE

    ELEMENT METHOD

    IVO BABUSKA, VICTOR NISTOR, AND NICOLAE TARFULEA

    Abstract. We propose a method for treating the Dirichlet boundary condi-tions in the framework of the Generalized Finite Element Method (GFEM).

    We are especially interested in boundary data with low regularity (possibly

    a distribution). We use approximate Dirichlet boundary conditions as in [11]and polynomial approximations of the boundary. Our sequence of GFEM-

    spaces considered, S, = 1, 2, . . . is such that S 6 H10 (), and hence itdoes not conform to one of the basic FEM conditions. Let h be the typ-ical size of the elements defining S and let u Hm+1() be the solutionof the Poisson problem u = f in , u = 0 on , on a smooth, boundeddomain . Assume that vH1/2() Ch

    m vH1() for all v S

    and |u uI |H1() Chm uHm+1(), u Hm+1() H10 (), for a suit-able uI S. Then we prove that we obtain quasi-optimal rates of con-vergence for the sequence u S of GFEM approximations of u, that is,u uH1() Chm fHm1(). We also extend our results to the inho-mogeneous Dirichlet boundary value problem u = f in , u = g on ,including the case when f = 0 and g has low regularity (i.e., is a distribu-tion). Finally, we indicate an effective technique for constructing sequences

    of GFEM-spaces satisfying our conditions using polynomial approximations ofthe boundary.

    Contents

    Introduction 2

    Part 1. Approximate Dirichlet boundary conditions 51. Homogeneous boundary conditions 52. Non-homogeneous boundary conditions 83. Distributional boundary data and the inf-sup condition 9

    Part 2. GFEM Approximation Spaces 114. The Generalized Finite Element Method 115. Properties of the spaces S 156. Interior numerical approximation 217. Comments and further problems 24References 24

    Date: April 30, 2007.I. Babuska was partially supported by NSF Grant DMS 0341982 and ONR Grant N00014-94-

    0401. V. Nistor was partially supported by NSF Grant DMS 0555831.

    1

  • 2 I BABUSKA, V. NISTOR, AND N. TARFULEA

    Introduction

    In the past few years, meshless methods for the approximation of solutions ofpartial differential equations have received increasing attention, especially in theEngineering and Physics communities. The reasons behind the development ofsuch methods are the difficulties associated to the mesh generation, particularlywhen the geometry of the domain is complicated. As in the case of the usual FiniteElement Method, one of the major problems in the implementation of meshlessmethods is the enforcement of Dirichlet boundary conditions. It is the purpose ofthis paper to address the problem of enforcing Dirichlet boundary conditions in theGeneralized Finite Element Method framework. We are especially interested in thecase when the Dirichlet boundary data has low regularity, including the case whenit is a distribution on the boundary.

    The classical Rayleigh-Ritz method for elliptic Dirichlet boundary value prob-lems requires that the trial subspace functions fulfill the boundary conditions. Nev-ertheless, the construction of such subspaces implies many difficulties in practicewhen the boundary of the domain is curved. Therefore, several approaches havebeen devised for dealing with the Dirichlet boundary conditions on domains withcurved boundaries. One approach is to modify the variational principles by addingappropriate boundary terms so that there will be no need for the trial subspaces tofulfill any condition at the boundary. See the works of Babuska [2, 4], Bramble andNitsche [12], and Bramble and Schatz [13, 14], among others, for examples of howthis approach works in practice. Another approach (used also in this paper) is touse subspaces with nearly zero boundary conditions. This ideea was first outlinedby Nitsche [29] and further studied by Berger, Scott, and Strang [11] and Nitsche[30].

    Yet another approach for dealing with the Dirichlet boundary conditions is theIsoparametric Finite Element Method or IFEM with curved finite elements alongthe boundary. See [18] and references therein, or [17, 19, 21, 23, 24, 35, 36], amongmany others, for more recent work and applications. This approach is typicallyused in connection with a numerical quadrature scheme computing the coefficientsof the resulting linear systems. In the applications of this method, except in specialcases (such as when is a polyhedral domain) the interior h of the union of thefinite elements is not equal to , although the boundary of h is very close to .That is, the approximate solution uh is sought in a subspace Vh H10 (h), and sothe homogeneous Dirichlet boundary condition u = 0 on is approximated bythe boundary condition uh = 0 on h. In fact, uh is the solution of a variationalequation ah(uh, vh) = (fh, vh)h for all vh Vh, where ah(, ) is a bilinear form whichapproximates the usual bilinear form defined over H1(h)H1(h), and fh V happroximates the linear form vh Vh

    hfvhdx, where f is an extension of f

    to the set h.Our approach has certain points in common with the isoparametric method just

    mentioned in the fact that we are using polynomial approximations of the boundary.However, our method does not require non-linear changes of coordinates. Ourmethod thus combines the approaches in the papers of Berger, Scott, and Strang[11] and Nitsche [30]. Our definition of the discrete solution is as in [11], whereas ourassumptions are closer to those of [30]. We have tried to keep our assumptions at aminimum. This is possible using partitions of unity, more precisely the Generalized

  • DIRICHLET PROBLEM 3

    Finite Element Method or GFEM, a method that originated in the work of Babuska,Caloz, and Osborn [6] and further developed in [4, 7, 8, 10, 22, 25, 27, 37].

    Our construction is different from the IFEM in that we do not require compli-cated non-linear changes of coordinates. Moreover, our method uses non-conformingsubspaces of functions and it does not have to deal with extensions over larger do-mains. It is closely related to [9] which uses GFEM for elliptic Neumann boundaryvalue problems with distributional boundary data. The GFEM is a generalizationof the meshless methods which use the idea of partition of unity. This methodallows a great flexibility in constructing the trial spaces, permits inclusion of apriori knowledge about the differential equation in the trial spaces, and gives theoption of constructing trial spaces of any desired regularity. We mention that theGFEM is also known and used under other names, such as: the method of clouds,the method of finite spheres, the Xfinite element method, and others. See thesurvey by Babuska, Banerjee, and Osborn [4] for further information and references.

    Let us now describe the main results of this paper in some detail. Let Rn bea smooth, bounded domain with boundary . Here are some of our assumptions.Let f L2() and u H2() be the unique solution of the Poisson problem

    (1) u = f on , u = 0 on .

    Assume that we are given a sequence h 0 and a sequence S H1() of test-trial spaces. The parameters h play the role of the size of the elements definingS. Let B(u, v) :=

    u vdx. We define the discrete solution u S in the

    usual way:

    (2) B(u, v) = f, vL2() :=

    fv dx, for all v S.

    We do not assume however that S satisfy exactly the Dirichlet boundary condi-tions, that is, we do not assume S H10 ().

    Let us fix from now on a natural number m N = {1, 2, . . .} that will play, inwhat follows, the role of the expected order of approximation. We shall make thefollowing two basic assumptions. The first assumption is that our approximatingfunctions satisfy Dirichlet boundary conditions approximately:

    Assumption 1, nearly zero boundary values: vH1/2() Chm vH1()for any v S, N.

    The second assumption is an approximation assumption that will be used also fornon-homogeneous boundary conditions. For that purpose, let us consider a secondsequence of subspaces S H1(), S S, which are not required to satisfy anyboundary conditions.

    Assumption 2, approximability: for any u Hm+1(), any 0 i j m + 1, and any N, there exists uI S such that |u uI |Hi() Chji uHj(). If u = 0 on , then we can take uI S.

    These two assumptions are formulated in more detail in Section 1.Our paper is divided into two parts. In the first part, we prove some general

    approximation results for the Poisson problem with Dirichlet boundary conditions.The approximations (or discrete solutions) belong to some abstract spaces S (forzero boundary conditions) or S (for general boundary conditions) that are re-quired to satisfy certain reasonable assumptions (Assumptions 1 and 2, for zeroDirichlet boundary conditions, and Assumptions 1, 2, and 3 for non-zero boundary

  • 4 I BABUSKA, V. NISTOR, AND N. TARFULEA

    conditions). In the second part of the paper we provide examples of GeneralizedFinite Element Spaces satisfying these assumptions and extend the results of thefirst part to boundary data g with low regularity. Under Assumptions 1 and 2, ourmain approximation result in Section 1 is the following.

    Theorem 0.1. Let S H1() be a sequence of finite dimensional subspacessatisfying Assumptions 1 and 2 for a sequence h 0 and 1 p m. Letf Hp1(). Then the (unique) solutions u and u of Equations (1) and (4)satisfy

    u uH1() ChpuHp+1() ChpfHp1(),for constants independent of N and f Hp1().

    In Section 2 we extend our results to the non-homogeneous Dirichlet boundaryconditions case u = g on , with g Hm+1/2(). In essence, we will be lookingfor a sequence Gk of approximate extensions of g, that is, a sequence of elements ofHm+1() subject to the following assumption. Recall that the sequence h shouldbe thought of as the typical size of the elements de

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